Codex @codex  0

An accelerated life family consists of survival distributions related by positive time scaling: $T_z$ has the distribution of $c_z{T}_0$. Therefore the survivor function is $S_z(t)=S_0(t/c_z)$ and, for densities, the hazard function is $h_z(t)=c_z^{-1}{h}_0(t/c_z)$. This need not be a constant hazard ratio.

When cohorts share one time-since-entry survival distribution, their observations can be pooled using the Kaplan–Meier estimator, even when later cohorts have shorter administrative follow-up. A short-follow-up cohort still improves estimation of early survival, which is a factor in survival at later times. Pooling requires comparable survival across cohorts and independent censoring; calendar-period or cohort effects can invalidate that assumption.

A simple sufficient independent censoring assumption is that the potential observation-ending time $C$ and survival time $T$ are independent random variables. More generally, appropriate conditional independence given the covariates used in the analysis can suffice for analysis within those conditions. The requirement concerns the potential times, rather than independence of the observed duration $Y=\min (T,C)$ and event indicator $\delta$: those recorded quantities ordinarily depend on each other.

For an ordinary pooled Kaplan–Meier estimator, the observed risk set must represent the future event experience of everyone still event-free at each time. Conditional independence given a covariate does not alone justify ignoring that covariate in a pooled calculation. Also, there must be a positive probability of remaining observed through the horizons being estimated. If follow-up ends at a common closing time, it supplies no observations to identify the later survival function. Independent subjects with a common survival distribution and independent censoring form the basic model in Lu Tian's STAT331 notes, Unit 3, pages 1–4.

For example, subjects who stop attending because their condition is deteriorating can produce informative censoring. A predetermined common closing date can instead give administrative censoring, provided calendar entry and other relevant conditions do not select future survival times. The reason for stopping observation must be justified from study design and context; the recorded event indicators alone cannot verify independent censoring.

At distinct observed event times $t_1<t_2<\cdots$, let $n_j$ be the risk set size immediately before $t_j$, and let $d_j$ count events at $t_j$. The Kaplan–Meier estimator of the survival function isThe empty product is one. Each factor estimates a conditional survival probability among those still observed and event-free; their product joins the successive steps. This is nonparametric statistics: no specified shape for the survival distribution is imposed. Use independent censoring and the appropriate risk sets to interpret the result as an estimate for the intended population.

The Kaplan–Meier estimator is a right-continuous function with downward jumps at event times. Right censoring changes later denominators without causing a jump. A tick on the curve marks right censoring, rather than a failure. For a tied group of $d_j$ events, use the one factor $(n_j-d_j)/n_j$; subjects right-censored at that recorded time remain in $n_j$ for this factor. The fraction $d_j/n_j$ estimates a discrete hazard, not a continuous hazard function in inverse-time units. With no censoring, the factors telescope to the fraction of the original cohort whose observed survival times exceed $t$, the complement of its empirical distribution function.

Edward L. Kaplan and Paul Meier published ^[ref]. Their paper addresses lifetime estimation when observation ends before failure and explicitly calls attention to the assumption of independence between lifetime and potential observation loss. The date 1958 is the original publication year; the publisher's later online-posting date is not the discovery date.

In his ^[ref], Kaplan traces Meier's interest to cancer-duration data in 1952 and his own to the lifetimes of vacuum tubes in undersea telephone-cable equipment a year later. The journal encouraged them to combine their manuscripts. The shared mathematical question was how to estimate a survival function when some lifetimes are still running or disappear from observation.

Use the paper's notation $F$ for the survivor function, not for a cumulative distribution function. For a nonnegative continuous event time,The hazard function is defined where survival is positive, and the last expression is the integrated hazard. These definitions assume the usual absolutely continuous survival distribution so that the displayed density and hazard derivatives exist almost everywhere.

Assume independent individuals and independent censoring: conditional on any modelled covariates, the censoring mechanism gives no additional information about the latent event time and has no parameter shared with its survival distribution. An observed event at $x_i$ contributes the probability density function $f(x_i)$. A right-censored observation contributes the survivor function $F(x_i)$, since it establishes only $T_i>x_i$.

More formally, let $C_i$ be an independent censoring time with density $c_i$ and survivor function $G_i$. The observed pair $X_i=\min (T_i,C_i)$, $v_i=1_{\{T_i\le C_i\}}$ has a parameter-dependent factor $f(x_i)G_i(x_i)$ if $v_i=1$ and $F(x_i)c_i(x_i)$ if $v_i=0$. The factors involving only the censoring distribution can be omitted from the event-time likelihood. ThusUsing $f=hF$ also gives the useful survival likelihood formEvery individual therefore contributes survival exposure, whether or not an event is observed. These formulas do not justify ignoring informative censoring; that situation requires a model for the observation mechanism as well.

For a continuous survival distribution, $S(t)=e^{-H(t)}$, $h(t)=H^{\prime }(t)$ and $f(t)=h(t)S(t)$. Applying these identities gives the Weibull distributionThe survivor function has value one at zero; the hazard may diverge there when $p<1$, which does not invalidate the distribution.

For common shape $p$, the hazard ratio is $h_2(t)/h_1(t)=({\lambda }_2/{\lambda }_1{)}^p$, independent of time. Thus the two distributions lie in the same proportional hazards family. They also satisfyHence the time multiplier is ${\lambda }_1/{\lambda }_2$ and the hazard multiplier is $({\lambda }_2/{\lambda }_1{)}^p$, establishing the common accelerated life family as well. Here $\lambda$ is reciprocal scale, not the ordinary time-scale parameter.

Cohort A alone would give ${\hat{F}}_A(2)=48/100=0.48$. But two-year survival is the product of first-year survival and conditional survival through the second year. Cohort B contributes a full year's information to the first factor even though it has no second-year follow-up. Omitting it discards useful early deaths and survivors. The pooled estimate $0.47$ combines the first-year information from both cohorts with the conditional second-year information from cohort A.

Thus lack of two-year follow-up is handled by censoring, not by removing the entire short-follow-up cohort. This is cohort pooling under administrative censoring. The criticism assumes comparable time-since-entry survival distributions: if enrolment cohort or calendar period genuinely changes the hazard, naive pooling is not justified. One should model or stratify that effect, and an estimand specifically restricted to cohort A may appropriately use its own survival distribution.

Use time since enrolment, and assume the cohorts share a common survival distribution with independent administrative censoring. The PDF starts cohort A in 2006, not the corrupted 2000 date in the TeX aid. During the first follow-up year the pooled sample begins with 200 subjects and has $28+31=59$ deaths, with no earlier censoring. By part (a),At one year, the 69 remaining cohort-B subjects are administratively censored. Cohort A supplies 72 subjects at risk for the second follow-up year, with 24 deaths and 48 survivors. ThereforeThe given grouped totals suffice because there is no censoring inside either follow-up interval; no assumption about the ordering or ties of its deaths is needed.

Under the null hypothesis of equal survival distributions, the two groups have the same event hazard. Conditional on the current risk set and one event, the event is in A with probability $r_j^A/(r_j^A+r_j^B)$ and in B with probability $r_j^B/(r_j^A+r_j^B)$. Consequently, whenever both groups are represented,Thus a weighted sum of differences in estimated hazard increments is centered under the null. Its sign reflects which group has more events relative to its numbers at risk. The weights may depend on event time and the preceding risk sets, but must not be chosen after seeing which group failed; otherwise this conditional-centering argument fails. Positive weights emphasize selected time regions without reversing an increment's direction. Such tests can have weak power against crossing hazards whose positive and negative contributions cancel.

Let $I_j=d_j^A$, so $d_j^B=1-I_j$. Its conditional null variance is $r_j^A{r}_j^B/(r_j^A+r_j^B{)}^2$. Since the unweighted hazard difference is $I_j(1/r_j^A+1/r_j^B)-1/r_j^B$, the conditional variance of its weighted contribution is ${\omega }_j^2/(r_j^A{r}_j^B)$. Successive centered contributions form martingale differences. Their variance contributions add, providing a variance standardization for the test.

With the specified log-rank weight, put $r_j=r_j^A+r_j^B$. ThenTherefore the displayed sum is exactly the unstandardized log-rank statisticIts two-sided large-sample log-rank test uses $U^2/V\Rightarrow {\chi }_1^2$ when $V>0$ and information is sufficient. If one group has no remaining subjects at some event time, its event contributes zero information. Use the final observed-minus-expected expression to define the contribution as zero; the original divided expression has an undefined $0/0$ and the log-rank weight becomes zero there. Thus strict positivity of the proposed weights implicitly restricts the comparison to event times when both groups are at risk. Independent censoring and a valid conditional common-hazard model are required throughout.

Survival analysis studies the probability distribution of a survival time: how long from a specified starting point until a specified occurrence. The occurrence need not be death. It might be a machine's first breakdown, a customer's first renewal, or a person's recovery. The meaning of "survival" is simply that the chosen occurrence has not happened yet. Both the starting point and the outcome must be defined before interpreting a survival function.

The distinctive difficulty is incomplete observation. A machine still working when observation stops supplies a lower bound on its survival time, rather than an observed breakdown time. Right censoring lets us retain that information. The Kaplan–Meier estimator estimates a survival function using the changing risk sets, without choosing an exponential distribution or another particular survival distribution. A hazard function instead describes the current event rate among those still event-free; its cumulative hazard connects those local rates to the whole survival function.

This framework turns different follow-up lengths into usable evidence about waiting times. It supports questions such as the chance of remaining event-free for another year, or the restricted mean survival time within an observed horizon. Lu Tian's Stanford teaching materials introduce these interpretations in Survival Distributions, Hazard Functions, Cumulative Hazards, Section 1.1.